# NCTM PROBLEM SOLVING STANDARDS

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NCTM problem solving standards by grade level

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Problem Solving Standard for Pre-K through Grade 2Instructional programs from prekindergarten through grade 12 should enable all students to— build new mathematical knowledge through problem solving;solve problems that arise in mathematics and in other contexts;apply and adapt a variety of appropriate strategies to solve problems;monitor and reflect on the process of mathematical problem solving.Problem solving is a hallmark of mathematical activity and a major means of developing mathematical knowledge. It is finding a way to reach a goal that is not immediately attainable. Problem solving is natural to young children because the world is new to them, and they exhibit curiosity, intelligence, and flexibility as they face new situations. The challenge at this level is to build on children's innate problem-solving inclinations and to preserve and encourage a disposition that values problem solving. Teachers should encourage students to use the new mathematics they are learning to develop a broad range of problem-solving strategies, to pose (formulate) challenging problems, and to learn to monitor and reflect on their own ideas in solving problems.What should problem solving look like in prekindergarten through grade 2? Problem solving in the early years should involve a variety of contexts, from problems related to daily routines to mathematical situations arising from stories. Students in the same classroom are likely to have very different mathematical understandings and skills; the same situation that is a problem for one student may elicit an automatic response from another. For instance, when first-grade students were working in small groups to create models of animals with geometric solids, some had difficulty seeing the parts of animals as geometric shapes. Other students readily saw that they could use seven rectangular prisms to make a giraffe (see figure 4.24). Similarly, the question "How many books would there be on the shelf if Marita put six books on it and Al put three more on it?" may not be a problem for the student who knows the basic number combination 6 and 3 and its connection with the question. For the student who has not yet learned the number combination and may not yet know how to represent the task symbolically, this problem presents an opportunity to learn the skills needed to solve similar problems. Fig. 4.24. A block giraffe made from seven rectangular prisms (Adapted from Russell, Clements, and Sarama [1998, p. 115])Solving problems gives students opportunities to use and extend their knowledge of concepts in each of the Content Standards. For example, many problems relate to classification, shape, or space: Which blocks will fit on this shelf? Will this puzzle piece fit in the space that remains? How are these figures alike and how are they different? In answering these questions, students are using spatial-visualization skills and their knowledge of transformations. Other problems support students' development of number sense and understanding of operations: How many more days until school vacation? There are 43 cards in this group; how many packets of 10 can we make? If there are 26 students in our class and 21 are here today, how many are absent? When young students solve problems that involve comparing and completing collections by using counting strategies, they develop a better understanding of addition and subtraction and the relationship between these operations.Posing problems, that is, generating new questions in a problem context, is a mathematical disposition that teachers should nurture and develop. Through asking questions and identifying what information is essential, students can organize their thoughts, as the following episode drawn from classroom observations demonstrates:Lei wanted to know all the ways to cover the yellow hexagon using pattern blocks. At first she worked with the blocks using fairly undirected trial and error. Gradually she became more methodical and placed the various arrangements in rows. The teacher showed her a pattern-block program on the class computer and how to "glue" the pattern-block designs together on the screen. Lei organized the arrangements by the numbers of blocks used and began predicting which attempts would be transformations of other arrangements even before she completed the hexagons (see fig. 4.25). The next challenge Lei set for herself was to see if she could create a hexagonal figure using only the orange squares. She had experimented with square blocks and could not make a hexagon. "But," she explained to her teacher, "it might be different on the computer," indicating that she felt the computer was a powerful problem-solving tool. Fig. 4.25. Organizing arrangements that make hexagonsKyle was certain that he could find more arrangements for hexagons than Lei had found. Other students joined the discussions between Lei and Kyle. When this activity created a great demand for "turns" with the pattern blocks and the computer, the teacher took advantage of the class's interest by having students discuss how they would know when an arrangement of blocks was a duplicate and how they might keep a written record of their work.Kyle's participation illustrates that students are persistent when problems are interesting and challenging. Their interest also stimulates curiosity in other students.Students working together often begin to solve problems one way and, before reaching a solution, change their strategies. In addition, as they create and modify their strategies, students often recognize the need to learn more mathematics. The following episode, drawn from classroom observation, illustrates how teachers can make a problem mathematically rich.Several first-grade classes in the same school were planting a garden in the school courtyard. The students wanted each class to have the same amount of space for planting; thus, how to divide the area into three equal parts was greatly debated. A walkway, two shade trees, and several benches complicated the discussions. The students began to list all their concerns and the questions they needed to answer before dividing the area for the garden: How big can the garden be? Do the three sections have to be the same shape? How can we be sure each class has the same amount of space?The teachers drew a large map for each class and indicated the approximate location and amount of space taken by trees, walkway, and benches. In one class the students decided they wanted rectangular gardens and needed to measure the courtyard to figure out how large the rectangles could be. After many measurements and much debate, they cut out three rectangles that were four feet by nine feet to show how big each garden could be. When they were not certain how to use this information on their map, the teacher showed them a scale on a road map and how map scales are used. She suggested appropriate dimensions for the three rectangles, which they cut and glued to their map.The second class began with a discussion of what "the same area" means. They used large-grid paper squares and taped them to their map to allocate the maximum space for gardening, counting carefully to be certain each class had the same number of squares even though the shapes of the regions were different. This group also needed to learn about scale to actually make a plan for marking off the gardens outside.Before voting on how to mark off the gardens, the two groups presented their plans to all the classes.Deciding how to share land for a garden is an example of a classroom-based problem that facilitates students' development of problem-solving strategies. The task was complex. The students struggled with how to share the area equally, how to measure, and how to communicate their ideas. However, the project was rich with proposed strategies, counterproposals, and opportunities for the teachers to introduce new mathematics.Children's literature is helpful in setting a context for both student-generated and teacher-posed problems. For example, after reading 1 Hunter (Hutchins 1982) to her class, a second-grade teacher asked students to figure out how many animals, including the hunter, were in the story. Figure 4.26 illustrates several approaches used by the students. Fig. 4.26. Detemining the number of animals in 1 Hunter Sharing gives students opportunities to hear new ideas and compare them with their own and to justify their thinking. As students struggle with problems, seeing a variety of successful solutions improves their » chance of learning useful strategies and allows them to determine if some strategies are more flexible and efficient. When the teacher invited the students to explain their solutions to the 1 Hunter problem, several of them discovered their counting or computational errors and made corrections during the presentations. Explaining their pictorial and written solutions helped them articulate their thinking and make it precise.What should be the teacher's role in developing problem solving in prekindergarten through grade 2? The decisions that teachers make about problem-solving opportunities influence the depth and breadth of students' mathematics learning. Teachers must be clear about the mathematics they want their students to accomplish as they structure situations that are both problematic and attainable for a wide range of students. They make important decisions about when to probe, when to give feedback that affirms what is correct and identifies what is incorrect, when to withhold comments and plan similar tasks, and when to use class discussions to advance the students' mathematical thinking. By allowing time for thinking, believing that young students can solve problems, listening carefully to their explanations, and structuring an environment that values the work that students do, teachers promote problem solving and help students make their strategies explicit. e-Example 4.3: Navigating Paths and Mazes (Part 2) Instead of teaching problem solving separately, teachers should embed problems in the mathematics-content curriculum. When teachers integrate problem solving into the context of mathematical situations, students recognize the usefulness of strategies. Teachers should choose specific problems because they are likely to prompt particular strategies and allow for the development of certain mathematical ideas. For example, the problem "I have pennies, dimes, and nickels in my pocket. If I take three coins out of my pocket, how much money could I have taken?" can help children learn to think and record their work.Assessing students' abilities to solve problems is more difficult than evaluating computational skills. However, it is imperative that teachers » gather evidence in a variety of ways, such as through students' work and conversations, and use that information to plan how to help individual students in a whole-class context. Knowing students' interests allows teachers to formulate problems that extend the mathematical thinking of some students and that also reinforce the concepts learned by other students who have not yet reached the same understandings. Classrooms in which students have ready access to materials such as counters, calculators, and computers and in which they are encouraged to use a wide variety of strategies support thinking that results in multiple levels of understanding. e-Example 4.4: Tangram Challenges (Part 2) Two examples illustrate how conversations with students give teachers useful information about students' thinking. Both examples have been drawn from observations of students.Katie, a kindergarten student, said that her sister in third grade had taught her to multiply. "Give me a problem," she said. The teacher asked, "How much is three times four?" There was a long pause before Katie replied, "Twelve!" When the teacher asked how she knew, Katie responded, "I counted ducks in my head—three groups with four ducks." Katie, while demonstrating an additive understanding of multiplication by counting the ducks in each group, was also exhibiting an interest in, and readiness for, mathematics that is traditionally a focus in the higher grades. Luis, a second grader, demonstrated fluency with composing and decomposing numbers when he announced that he could figure out multiplication. His teacher asked, "Can you tell me four times seven?" Luis was quiet for a few moments, and then he gave the answer twenty-eight. When the teacher asked how he got twenty-eight, Luis replied, "Seven plus three is ten, and four more is fourteen; six more is twenty and one more is twenty-one; seven more is twenty-eight." Luis's approach also built on additive thinking but with a far more sophisticated use of number relationships. He added 7 + 7 + 7 + 7 mentally by breaking the sevens into parts to complete tens along the way.Students are intrigued with calculators and computers and can be challenged by the mathematics that technology makes available to them, as shown in the following episode, adapted from Riedesel (1980, pp. 74–75):Erik, a very capable kindergarten student, observed his teacher using a calculator and asked how it worked. The teacher showed him how to compute simple additions. Erik took the calculator to the math corner and a few minutes later loudly proclaimed, "Five plus four equals nine. Hey, this thing got it right!" A few minutes later, he walked over to the teacher and they had the following conversation:Erik: What does this button mean?Teacher: That's called the "square root." It's a pretty difficult idea in math.Erik: OK. (He wanders away, but not for long.) But this is a disaster! I pressed 2, then the square-root key, and I got a whole lot of numbers.Teacher: Try using 1. (Erik tries this.) Erik: That just gives 1 back.Teacher: Try 4. (Erik notes that the result is 2 and asks why. The teacher tells him to get the square tiles and put out one.) Is that a square? (Erik nods.) Try to add more tiles beside this one until it is a square again. (Erik adds one tile.) Is that a square?Erik: No, it's a rectangle. (The teacher asks how he could make it into a square, and Erik adds two more tiles.) Teacher: How many tiles are there in all? (Erik responds that there are four.) Good. Press 4 on the calculator. How long is the bottom of the square?Erik: Two.Teacher: And here at the left side?Erik: Two there, too.Teacher: Press the square-root key.Erik: Hey, it comes out 2!The teacher challenged Erik to add more tiles until he made another square. Erik built a 33 array, counted the total tiles, entered this number into the calculator, and pressed the square-root key. He found that the result was the number of rows and also the number of tiles in each row. Erik kept building squares until at a 99 array he said his eyes hurt. The teacher asked him what he had found out.Erik: Well, if you make a square, then all you have to do is count the tiles and press that number and the square-root key and the calculator tells you how many tiles there are on each side.Teacher: Good work! What else does that number mean?Erik: It means that there are that many rows and that many tiles in each row. (The teacher congratulates Erik on figuring this out.) Yeah, I guess if you want to learn something really bad, you can. Tomorrow, I'm going to go up to one hundred!Teachers should ask students to reflect on, explain, and justify their answers so that problem solving both leads to and confirms students' understanding of mathematical concepts. For example, following an estimation activity in a first-grade class, students learned that there were eighty-three marbles in a jar. There were twenty-five students in the class, so the teacher asked how many marbles each child could get. Graham said, "Three." When the teacher asked how he knew, Graham replied, "Eighty-three is just a little more than seventy-five, so we only get three. There are four quarters in a dollar. There are three quarters in seventy-five cents. So we can only get three."Teachers must make certain that problem solving is not reserved for older students or those who have "got the basics." Young students can engage in substantive problem solving and in doing so develop basic skills, higher-order-thinking skills, and problem-solving strategies (Cobb et al. 1991; Trafton and Hartman 1997).

Pre K-2

Problem Solving Standard for Grades 3-5Instructional programs from prekindergarten through grade 12 should enable all students to— build new mathematical knowledge through problem solving;solve problems that arise in mathematics and in other contexts;apply and adapt a variety of appropriate strategies to solve problems;monitor and reflect on the process of mathematical problem solving.Problem solving is the cornerstone of school mathematics. Without the ability to solve problems, the usefulness and power of mathematical ideas, knowledge, and skills are severely limited. Students who can efficiently and accurately multiply but who cannot identify situations that call for multiplication are not well prepared. Students who can both develop and carry out a plan to solve a mathematical problem are exhibiting knowledge that is much deeper and more useful than simply carrying out a computation. Unless students can solve problems, the facts, concepts, and procedures they know are of little use. The goal of school mathematics should be for all students to become increasingly able and willing to engage with and solve problems.Problem solving is also important because it can serve as a vehicle for learning new mathematical ideas and skills (Schroeder and Lester 1989). A problem-centered approach to teaching mathematics uses interesting and well-selected problems to launch mathematical lessons and engage students. In this way, new ideas, techniques, and mathematical relationships emerge and become the focus of discussion. Good problems can inspire the exploration of important mathematical ideas, nurture persistence, and reinforce the need to understand and use various strategies, mathematical properties, and relationships.What should problem solving look like in grades 3 through 5? Students in grades 3–5 should have frequent experiences with problems that interest, challenge, and engage them in thinking about important mathematics. Problem solving is not a distinct topic, but a process that should permeate the study of mathematics and provide a context in which concepts and skills are learned. For instance, in the following hypothetical example, a teacher poses these questions to her students: If you roll two number cubes (both with the numbers 1–6 on their faces) and subtract the smaller number from the larger or subtract one number from the other if they are the same, what are the possible outcomes? If you did this twenty times and created a chart and line plot of the results, what do you think the line plot would look like? Is one particular difference more likely than any other differences?Initially, the students predict that they will roll as many of one difference as of another. As they begin rolling the cubes and making a list of the differences, some are surprised that the numbers in their lists range only from 0 to 5. They organize their results in a chart and continue to mark the differences they roll (see fig. 5.24). After the students have worked for a few minutes, the teacher calls for a class discussion and asks the students to summarize their results and reflect on their predictions. Some notice that they are getting only a few 0s and 5s but many 1s and 2s. This prompts the class to generate a list of rolls that produce each difference. Others list combinations that produce a difference of 2 and find many possibilities. The teacher helps students express this probability and questions them about the likelihood of rolling other differences, such as 0, 3, and 5. Fig. 5.24. A chart of the frequency of the differences between the numbers on the faces of two dice rolled simultaneously The questions posed in this episode were "problems" for the students in that the answers were not immediately obvious. They had to generate and organize information and then evaluate and explain the results. The teacher was able to introduce notions of probability such as predicting and describing the likelihood of an event, and the problem was accessible and engaging for every student. It also provided a context for encouraging students to formulate a new set of questions. For example: Could we create a table that would make it easy to compute the probabilities of each value? Suppose we use a set of number cubes with the numbers 4–9 on the faces. How will the results be similar? How will they be different? What if we change the rules to allow for negative numbers?Good problems and problem-solving tasks encourage reflection and communication and can emerge from the students' environment or from purely mathematical contexts. They generally serve multiple purposes, such as challenging students to develop and apply strategies, introducing them to new concepts, and providing a context for using skills. They should lead somewhere, mathematically. In the following episode drawn from an unpublished classroom experience, a fourth-grade teacher asked students to work on the following task:Show all the rectangular regions you can make using 24 tiles(1-inch squares). You need to use all the tiles. Count and keep a record of the area and perimeter of each rectangle and then look for and describe any relationships you notice.When the students were ready to discuss their results, the teacher asked if anyone had a rectangle with a length of 1, of 2, of 3, and so on, and modeled a way to organize the information (see fig. 5.25).The teacher asked if anyone had tried to form a rectangle of length 5 and, if so, what had happened. The students were encouraged work with partners to make observations about the information in the chart and their rectangular models. They noticed that the numbers in the first two columns of any row could be multiplied to get 24 (the area). The teacher noted their observation by writing "L x W = 24" and used the term factors of 24 as another way, in addition to length and width, to describe the numbers in the first two columns. Some students noticed that as the numbers for one dimension increased, those for the other dimension decreased. Still others noted that the perimeters were always even. One student asked if the rectangles at the bottom of the chart were the same as the ones at the top, just turned different ways. This observation prompted the teacher to remind the students that they had talked about this idea as a property of multiplication—the commutative property—and as congruence of figures.The teacher then asked the students to describe the rectangles with the greatest and smallest perimeters. They pointed out that the long "skinny" rectangles had greater perimeters than the "fatter" rectangles. The teacher modeled this by taking the 1-unit-by-24-unit rectangle of perimeter 50, splitting it in half, and connecting the halves to form the 2-unit-by-12-unit rectangle (see fig. 5.26). As she moved the tiles, she explained that some tile edges on the outside boundary of the skinny rectangle were moved to the inside of the wider rectangle. Because there were fewer edges on the outside, the perimeter of the rectangle decreased. Fig. 5.25. The dimensions of the rectangular regions made with 24 one-inch square tiles Fig. 5.26. Forming a 2 x 12 rectangle from a 1 x 24 rectangleThe "24 tiles" problem provides opportunities for students to consider the relationship between area and perimeter, to model the commulative property of multiplication, to use particular vocabulary (factor and multiple), to record data in an organized way, and to review basic number combinations. It reinforces the relationship L x W = A. It also allows the teacher to help students with different needs focus on different aspects of the problem—building all the rectangles, organizing the data, looking for patterns, or making and justifying conjectures.Reflecting on different ways of thinking about and representing a problem solution allows comparisons of strategies and consideration of different representations. For example, students might be asked to find several ways to determine the number of dots on the boundary of the square in figure 5.27 and then to represent their solutions as equations (Burns and Mclaughlin 1990). Fig 5.27. The "dot square" problemStudents will likely see different patterns. Several possibilities are shown in figure 5.28. The teacher should ask each student to relate the drawings to the numbers in their equations. When several different strategies have been presented, the teacher can ask students to examine the various ways of solving the problem and to notice how they are alike and how they are different. This problem offers a natural way to introduce the concept and term equivalent expressions. In addition to developing and using a variety of strategies, students also need to learn how to ask questions that extend problems. In this way, they can be encouraged to follow up on their genuine curiosity about mathematical ideas. For example, the teacher might ask students to create a problem similar to the "dot square" problem or to extend it in some way: If there were a total of 76 dots, how many would be on each side of the square? Could a square be formed with a total of 75 dots? Students could also work with extensions involving dots on the perimeter of other regular polygons. By extending problems and asking different questions, students become problem posers as well as problem solvers. Fig. 5.28. Several possible solutions to the "dot square" problemWhat should be the teacher's role in developing problem solving in grades 3 through 5? Teachers can help students become problem solvers by selecting rich and appropriate problems, orchestrating their use, and assessing students' understanding and use of strategies. Students are more likely to develop confidence and self-assurance as problem solvers in classrooms where they play a role in establishing the classroom norms and where everyone's ideas are respected and valued. These attitudes are essential if students are expected to make sense of mathematics and to take intellectual risks by raising questions, formulating conjectures, and offering mathematical arguments. Since good problems challenge students to think, students will often struggle to arrive at solutions. It is the teacher's responsibility to know when students need assistance and when they are able to continue working productively without help. It is essential that students have time to explore problems. Giving help too soon can deprive them of the opportunity to make mathematical discoveries. Students need to know that a challenging problem will take some time and that perseverance is an important aspect of the problem-solving process and of doing mathematics.As students share their solutions with classmates, teachers can help them probe various aspects of their strategies. Explanations that are simply procedural descriptions or summaries should give way to mathematical arguments. In this upper elementary class, a teacher questioned two students as they described how they divided nine brownies equally among eight people (Kazemi 1998, pp. 411–12):Sarah: The first four we cut them in half. (Jasmine divides squares in half on an overhead transparency.) Ms. Carter: Now as you explain, could you explain why you did it in half?Sarah: Because when you put it in half, it becomes four ... four ... eight halves.Ms. Carter: Eight halves. What does that mean if there are eight halves?Sarah: Then each person gets a half.Ms. Carter: Okay, that each person gets a half. (Jasmine labels halves1 through 8 for each of the eight people.) Sarah: Then there were five boxes [brownies] left. We put them in eighths.Ms. Carter: Okay, so they divided them into eighths. Could you tell us why you chose eighths?Sarah: It's easiest. Because then everyone will get ... each person will get a half and (addresses Jasmine)... how many eighths?Jasmine: (Quietly) Five-eighths.Ms. Carter: I didn't know why you did it in eighths. That's the reason. I just wanted to know why you chose eighths.Jasmine: We did eighths because then if we did eighths, each person would get each eighth, I mean one-eighth out of each brownie.Ms. Carter: Okay, one-eighth out of each brownie. Can you just, you don't have to number, but just show us what you mean by that? I heard the words, but...Jasmine: (Shades in one-eighth of each of the five brownies that were divided into eighths.)Person one would get this ... (points to one-eighth) Ms. Carter: Oh, out of each brownie.Sarah: Out of each brownie, one person will get one-eighth.Ms. Carter: One-eighth. Okay. So how much then did they get if they got their fair share?Jasmine & Sarah: They got a half and five-eighths.Ms. Carter: Do you want to write that down at the top, so I can see what you did? (Jasmine writes 1/2 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 at the top of the overhead transparency.) In this discussion, the teacher pressed students to give reasons for their decisions and actions: What does it mean if there are eight halves? Could you tell us why you chose eighths? Can you show us what you mean by that? She was not satisfied with a simple summary of the steps but instead expected the students to give verbal justifications all along the way and to connect those justifications with both numbers and representations. This particular pair of students used a strategy that was different from that of other students. Although it was not the most efficient strategy, it did reveal that these students could solve a problem they had not encountered before and that they could explain and represent their thinking.Listening to discussions, the teacher is able to assess students' understanding. In the conversation about sharing brownies, the teacher asked students to justify their responses in order to gain information about their conceptual knowledge. For any assessment of problem solving, teachers must look beyond the answer to the reasoning behind the solution. This evidence can be found in written and oral explanations, drawings, and models. Reflecting on these assessment data, teachers can choose directions for future instruction that fit with their mathematical goals.

3rd-5th

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